﻿ 基于改进McEvily模型的结构压-压疲劳寿命预测方法
 舰船科学技术  2018, Vol. 40 Issue (12): 57-63 PDF

Compression to compression fatigue life analysis of structures based on extended McEvily model
XU Fei-ran, LUO Guang-en, SHEN Yan
School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: In the diving process, most parts of the submersible suffer from compressive stress. The pressure changes with the changeable diving depth, which will likely lead to fatigue damage of the spherical shell. It is very necessary to study the fatigue problem under the cyclic compressive loading because it is quite different from the conventional fatigue analysis in mainly tensile stress state. Based on the theory of fracture mechanics, the extended McEvily crack growth rate model is adopted to predict the fatigue life of deep sea structures under compressive cyclic loading. Firstly, with the finite element method, the structure is simulated and the influence of finite element size in the crack tip region is investigated. Secondly, the multi-step analysis combined with the node release technique is used to calculate the residual tensile stress and stress intensity factor along the crack plane under cyclic compressive loading. Then, the crack growth life curve is obtained by the extended McEvily model. Finally, taking a two sided cracked plate under unidirectional cyclic compression for example to illustrate the calculation method and make a comparison between the calculation results and the experiment data. It turns out that the method is feasible and effective, which can provide references for the fatigue life evaluation of structures under cyclic compressive loading.
Key words: extended McEvily model     compressive cyclic loading     fatigue life     numerical simulation
0 引　言

1 疲劳寿命预测方法

1.1 改进的McEvily疲劳裂纹扩展模型

 $\frac{{{\rm{d}}a}}{{{\rm{d}}N}} = \frac{{A{{\left[ {{K_{\max }} \cdot \left( {1 - {f_{{\rm{op}}}}} \right) - \Delta {K_{{\rm{effth}}}}} \right]}^m}}}{{1 - {{\left( {{K_{\max }}/{K_C}} \right)}^n}}}{\text{。}}$ (1)

 $\left\{ \begin{array}{l}{K_{\max }} = \sqrt {{\text{π}} {r_e}\left( {{\rm Sec}\displaystyle\frac{{{\text{π}} \left| {{\sigma _{\max }}} \right|}}{{2{\sigma _{\rm{V}}}}} + 1} \right)} \left[ {1 + Y\left( a \right)\sqrt {\displaystyle\frac{a}{{2{r_e}}}} } \right]{\sigma _{\max }},\\{K_{\min }} = \sqrt {{\text{π}} {r_e}\left( {{\rm Sec}\displaystyle\frac{{{\text{π}} \left| {{\sigma _{\min }}} \right|}}{{2{\sigma _{\rm{V}}}}} + 1} \right)} \left[ {1 + Y\left( a \right)\sqrt {\displaystyle\frac{a}{{2{r_e}}}} } \right]{\sigma _{\min }},\\\Delta K = {K_{\max }} - {K_{\min{\text{。}} }}\end{array} \right.$
 ${f_{{\rm{op}}}} = \left\{ \begin{array}{l}\max \left\{ {R,{A_0} + {A_1}R + {A_2}{R^2} + {A_2}{R^3}} \right\},\;\;\;\;0 \leqslant R < 1,\\{A_0} + {A_1}R,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - 2 \leqslant R < 0{\text{。}}\end{array} \right.$
 $\left\{ \begin{array}{l}{A_0} = \left( {0.825 - 0.34\alpha ' + 0.05{{\alpha '}^2}} \right) \cdot {\left[ {\cos \left( {{\text{π}} {\sigma _{\max }}/2{\sigma _{{\rm{fl}}}}} \right)} \right]^{1/\alpha '}},\\{A_1} = \left( {0.415 - 0.071\alpha '} \right) \cdot {\sigma _{\max }}/{\sigma _{{\rm{fl}}}},\\{A_2} = 1 - {A_0} - {A_1} - {A_3},\\{A_3} = 2{A_0} + {A_1} - 1,\\{\sigma _{{\rm{fl}}}} = \left( {{\sigma _Y} + {\sigma _u}} \right)/2,\\\alpha ' = \displaystyle\frac{1}{{1 - 2v}} + \displaystyle\frac{{1 - \frac{1}{{1 - 2v}}}}{{{{\left[ {1 + 0.886\ 1 \cdot {{\left( {t/{{\left( {{K_{\max }}/{\sigma _{\rm{Y}}}} \right)}^2}} \right)}^{3.2251}}} \right]}^{0.75952}}}}{\text{。}}\end{array} \right.$
 $\left\{\!\!\! \begin{array}{l}{K_C} = \left[ {\displaystyle\frac{{{{\left( {1 - 2v} \right)}^2} - \sqrt {1 - {v^2}} }}{{{{\left( {1 - 2v} \right)}^2} - 1}} \cdot \frac{{{\text{π}} \cdot \lambda }}{{{{\left( {1 - 2v} \right)}^2}}} + \frac{{\sqrt {1 - {v^2}} - 1}}{{{{\left( {1 - 2v} \right)}^2} - 1}}} \right] \cdot {K_{{\rm{IC}}}},\\\lambda = \displaystyle\frac{{{{\left( {1 - 1.65v} \right)}^2}}}{5} - \frac{1}{{20n'}}{\left[ {{{\left( {1 - 1.65v} \right)}^2}} \right]^{\frac{1}{{n'}}}} +\\ \displaystyle\frac{{\frac{1}{{\text{π}} } - \frac{1}{{2.2n'}}{{\left( {\frac{1}{{\text{π}} }} \right)}^{\frac{1}{{n'}}}} - \left[ {\frac{{{{\left( {1 - 1.65v} \right)}^2}}}{5} - \frac{1}{{20n'}}{{\left( {{{\left( {1 - 1.65v} \right)}^2}} \right)}^{\frac{1}{{n'}}}}} \right]}}{{{{\left[ {1 + \frac{{t/{{\left( {{K_{\max }}/{\sigma _Y}} \right)}^2}}}{{1 + 1/n'}}} \right]}^{1.6 + 1/n'}}}}{\text{。}}\end{array} \right.$
 $\frac{{{\sigma _v}}}{{{\sigma _u}}} = \frac{{\text{π}} }{2} \cdot \frac{1}{{{{\cos }^{ - 1}}\left( {\frac{1}{{{\beta ^2} - 1}}} \right)}},\beta = \frac{{{K_{_C}}}}{{{\sigma _u}\sqrt {{\text{π}} {r_e}\left( {1 + \frac{{Y\left( {{r_e}} \right)}}{{\sqrt 2 }}} \right)} }} > \sqrt 2 {\text{。}}$

1.2 应力强度因子计算

 ${K_{res}}\left( {linear} \right) = 2\sqrt {a/{\text{π}}} \int_0^a {\frac{{{\sigma _{res}}\left( x \right)}}{{\sqrt {{a^2} - {x^2}} }}} {\text{。}}$ (2)

 ${\bar \sigma _{res}} = {K_{res}}\left( {linear} \right)/\left[ {Y\left( a \right)\sqrt {{\text{π}}a} } \right],$ (3)
 ${K_{res}} = \sqrt {{\text{π}} {r_e}\left( {{\rm Sec}\frac{{\text{π}} }{2}\frac{{{{\bar \sigma }_{res}}}}{{{\sigma _v}}} + 1} \right)} \left( {1 + Y\left( a \right)\sqrt {\frac{a}{{2{r_e}}}} } \right){\bar \sigma _{res}}{\text{。}}$ (4)
2 算例及验证 2.1 压-压疲劳试验

 图 2 裂纹扩展速率曲线 Fig. 2 Crack growth rate curve
2.2 裂纹扩展速率模型

 $\frac{{{\rm{d}}a}}{{{\rm{d}}N}} = \frac{{1.513 \times {{10}^{ - 11}} \cdot {{\left[ {{K_{\max }} \cdot (1 - {f_{{\rm{op}}}}) - 2.83} \right]}^{2.791}}}}{{1 - {{({K_{\max }}/150)}^6}}}{\text{。}}$ (5)

 $\frac{{{\rm{d}}a}}{{{\rm{d}}N}} = \frac{{1.513 \times {{10}^{ - 11}} \cdot {{\left[ {{K_{\max }} \cdot (1 - 0) - 0} \right]}^{2.791}}}}{{1 - {{({K_{\max }}/150)}^6}}}{\text{。}}$ (6)
2.3 残余应力有限元分析和疲劳寿命计算 2.3.1 有限元模型的建立

 图 3 材料应力应变曲线 Fig. 3 Stress-strain curves of materials

 图 4 试件有限元网格划分 Fig. 4 Finite element mesh generation of specime

 图 5 接触面定义及边界条件 Fig. 5 Contact surface and boundary condition
2.3.2 有限元单元尺寸影响

Solanki[13]研究了平面应力和平面应变状态下CT、MT模型，指出裂纹尖端反复塑性区范围内至少包含3～4个单元，Newman[14]将塑性区单元数从10改进为20，以获取更为准确的计算结果。

 图 6 最大载荷时裂纹尖端塑性区 Fig. 6 Crack tip plastic zone under maximum load

 图 8 网格细化尺寸-线性应力强度因子关系 Fig. 8 Relationship between mesh size and linear stress intensity factor

2.3.3 多载荷步结合节点释放分析方法

 图 9 直接计算法对比载荷步结合节点释放方法 Fig. 9 Comparison of node release method and direct calculation method
2.4 结果对比分析

 图 10 a=20 mm时的残余应力分布 Fig. 10 Residual stress distribution at a=20 mm

 图 11 a=21 mm时的残余应力分布 Fig. 11 Residual stress distribution at a=21 mm

 图 12 a=22 mm时的残余应力分布 Fig. 12 Residual stress distribution at a=22 mm

 图 13 a=23 mm时的残余应力分布 Fig. 13 Residual stress distribution at a=23 mm

 图 14 a=24 mm时的残余应力分布 Fig. 14 Residual stress distribution at a=24 mm

 图 15 a=25 mm时的残余应力分布 Fig. 15 Residual stress distribution at a=25 mm

 图 16 a=26 mm时的残余应力分布 Fig. 16 Residual stress distribution at a=26 mm

 图 17 非线性应力强度因子拟合曲线-裂纹长度关系 Fig. 17 Relationship between nonlinear stress intensity factor fitting curve and crack length

 图 18 预测的a-N曲线与试验结果对比 Fig. 18 Comparison of predicted a-N curves and experiment results

3 结　语

1）由于材料塑性，压缩循环载荷会在裂纹尖端区域产生残余拉伸应力，它是压-压载荷下疲劳裂纹扩展的主要驱动力。

2）裂纹尖端区域有限元单元尺寸对塑性引起的残余应力和应力强度因子的计算结果有较大影响，本文研究了一系列不同单元尺寸对计算结果的影响，综合考虑计算精度和计算时间，建议裂尖区域单元尺寸值取为 ${d_p}/40$ ，即裂尖最大塑性区尺寸 ${d_p}$ 的1/40。

3）多载荷步结合节点释放技术能较好模拟材料塑性对于每个载荷循环的影响，能够较为准确地计算沿着裂纹的残余应力分布。

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